The Alternating Projection Algorithm for the Symmetric Arrowhead Solution of Matrix Equation A×B=C

[1]

**Minghui Wang**, Department of Mathematics, Qingdao University of Science & Technology, Qingdao, China.

[2]

**Luping Xu**, Department of Mathematics, Qingdao University of Science & Technology, Qingdao, China.

[3]

**Juntao Zhang Zhang**, Department of Mathematics, Qingdao University of Science & Technology, Qingdao, China.

Based on the alternating projection (AP) algorithm, the constrained matrix equation A×B=C and the associate optimal approximation problem are considered for the symmetric arrowhead matrix solutions in the premise of consistency. The convergence results of the method are presented. At last, a numerical example is given to illustrate the efficiency of this method.

Iterative Method, Symmetric Arrowhead Matrix, The Alternating Projection Algorithm

[1]

Y. F. Xu, An inverse eig-envalue problem for a special kind of matrices. Math. Appl., 1 (1996) 68-75.

[2]

C. J. Meng, X. Y. Hu, L. Zhang, The skew symmetric orthogonal solution of the matrix equation AX=B, Linear Algebra Appl. 402 (2005) 303-318.

[3]

Li Jiaofen, Zhang Xiaoning, Peng Zhenyun, Alternative projection algorithm for single variable linear constraints matrix equation problems, Mathematica Numerical Since, 36 (2014) 143-162.

[4]

Y. X. Peng, X. Y. Hu, L. Zhang, An interation method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB = C, Applied Mathematics and Computation, 160 (2005) 763-777.

[5]

Z. Y. Peng, A matrix LSQR iterative method to solve matrix equation AXB = C, International Journal of Computer Mathematics, 87 (2010) 1820-1830.

[6]

J. F. Li, X. F. Duan, L. Zhang, Numerical solutions of AXB = C for mirror symmetric matrix X under a specified submatrix constraint. Computing, 90 (2010) 39-56.

[7]

Von Neumann J., Functional Operators. II. The Geometry of Spaces, Annals of Mathematics Studies, vol.22, Princeton University Press, Princeton, 1950.

[8]

Cheney W., Goldstein A. Proximity maps for convex sets, Proceedings of the American Mathematical Society, 10 (1959) 448-450.

[9]

Hongyi Li, Zongsheng Gao, Di Zhao. Least squares solutions of the matrix equation with the least norm for symmetric arrowhead matrices. Appl. Math. Comput., 226(2014) 719-724.

[10]

Y. F. Xu. An inverse eigenvalue problem for a special kind of matrices. Math. Appl., 1(1996)68-75.

[11]

G. P. Xu, M. S. Wei, D. S. Zhang, On solutions of matrix equations AXB +CYD=E. Linear Algebra Appl., 279 (1998) 93-109.

[12]

A. P. Liao, Z. Z. Bai, Y. Lei, Best approximate solution of matrix equation AXB +CYD=E. SIAM J. Matrix Anal. Appl., 27(2006) 675-688.

[13]

Lin T Q. Implicitly restarted global FOM and GMRES for nonsymmetric matrix equations and Sylvester equations[J]. Appl. Math. Comput., 167 (2005) 1004-1025.

[14]

Shifang Yuan, Qingwen Wang and Xuefeng Duan, On solutions of the quaternion matrix equation AX=B and their application in color image restoration [J], Appl. Math. Comput., 221 (2013) 10-20.

[15]

Yuming Feng, Chuandong Li, Tingwen Huang. Sandwich control systems with impulse time windows [J]. International Journal of Machine Learning and Cybernetics. (2016) 1-7.

[16]

Yuming Feng, Chuandong Li, Tingwen Huang. Periodically multiple state-jumps impulsive control systems with impulse time windows, Neurocomputing 193 (2016) 7–13.

[17]

Huamin Wang, Shukai Duan, Chuandong Li, Lidan Wang, Tingwen Huang. Linear impulsive control system with impulse time windows, Journal of Vibration and Control, 14 (2016) 1-8.